Suppose we have an ordinal, $a$, that is defined in the standard von Neumann way. Can we create a binary operation $+:a^2\to a$, such that $(a,+)$ is an Abelian group? Specifically, I would like $+$ to be defined with standard set-theoretical means. One solution I thought of was to define a function, $S:a\to a$, such that $$ S(x)= \begin{cases} \emptyset&\text{if } x = \sup(a)\\ x\cup \{x\}&\text{else} \end{cases}$$ From this, we could then define $S^{[y]}(x)$ as the $y$th composition of $S$ with itself. Specifically that $$S^{[y]}(x)=(\underset{i\in y}{C} S)(x)$$ (I assume that $S^{[y]}(x)=S^{[x]}(y)$ but I do not know how to prove this, so I may be mistaken.) From this we can define $x+y:= S^{[x]}(y)$. We can see that all group axioms are satisfied here. There is an additive identity ($0$), there also exist inverses ($(a-x) + x=0$), and the operation is closed.
But then, it seems that all groups formed are cyclic, and then the group made from $\omega_0$ would be isomorphic to the one from $\omega_1$, by virtue of $\mathbb{Z}$ being the only infinite cyclic group, thus the cardinality of the $\omega_1$ group would be equal to $\aleph_0$, which is smaller than the cardinality of $\omega_1$. This seems paradoxical, so I assume there is a flaw in my construction. What is the flaw, and how would I go about properly constructing a group from an ordinal?
Edit: In the comments, I've been informed that what I have is not guaranteed to be a group. I was confused by the existence of $\omega_0+1$ and thought that there also existed $\omega_0-1$, which is false. In which case, the problem remains of how to construct a group from an ordinal.
Edit 2: From the comments, it seems like there, in fact, is a group operation over every ordinal. But the question remains of is such a group operation definable in a "natural" way, using only basic set operations such as union and intersection. Ideally, the group operation should also be such that if $a$ and $b$ are ordinals, $(a,+_a)\cong (b,+_b)$ if and only if $|a| = |b|$. The goal here is to be able to construct groups of arbitrary size.